Optical lattice experiments at unobserved conditions with generative adversarial deep learning

Optical lattice experiments with ultracold atoms allow for the experimental realization of contemporary problems in many-body physics. Yet, devising models that faithfully describe experimental observables is often difficult and problem dependent; there is currently no theoretical method which accounts for all experimental observations. Leveraging the large data volume and presence of strong correlations, machine learning provides a novel avenue for the study of such systems. It has recently been proven successful in analyzing properties of experimental data of ultracold quantum gases. Here we show that generative deep learning succeeds in the challenging task of modeling such an experimental data distribution. Our method is able to produce synthetic experimental snapshots of a doped two-dimensional Fermi-Hubbard model that are indistinguishable from previously reported experimental realizations. We demonstrate how our generative model interprets physical conditions such as temperature at the level of individual configurations. We use our approach to predict snapshots at conditions and scales which are currently experimentally inaccessible, mapping the large-scale behavior of optical lattices at unseen conditions.

Figure 1

(a) Experimental realization of the Fermi-Hubbard model with ultracold atoms in a two-dimensional square optical lattice. (b) Our generative model is trained on site-resolved snapshots of quantum states, obtained at a fixed temperature and for a range of hole doping values $\delta$. (c) Sign-corrected spin correlations $C_s\left(r\right)$ of Eq. (2) as a function of the hole doping $\delta$ for nearest neighbors ($r=1$). The correlations obtained with our generative model “RUGAN” (blue squares, 1000 snapshots generated for each doping as to obtain statistics similar to the experiment) are consistent with experiment (orange circles). We also show results obtained through sprinkled holes (dashed line) and geometric string theory (solid line). Our generative model is not optimized on data corresponding to $\delta \gtrsim 0.24$ (shaded area). The calculation of this observable is described in Appendix pp4. (d) Same as in panel (c) but now for next-nearest neighbors ($r=\sqrt{2}$).

Figure 2

(a) Through a series of residual convolutional layers, the generative neural network transforms a latent sample to a new snapshot of a single spin species at a prescribed value of the doping $\delta$. (b) The distribution of snapshots created by the RUGAN (blue, color online) cannot be distinguished from the experimental snapshots (orange) with other unsupervised machine learning methods. Here, we show a dimensionality reduction with the UMAP algorithm [25] of an equal number of synthetic and experimental snapshots for three different values of $\delta$. (c) The number of strings exceeding a length of 2 per system site and (d) the average length (sites) of the string patterns detected by the algorithm described in Ref. [26], as a function of the doping $\delta$. The string statistics obtained with RUGAN (with which 1000 snapshots were created for each doping value) are consistent with experimental observations. The intermediate doping values $0.09\lesssim \delta \lesssim 0.23$ in the shaded area are not included during training of the generative model. (Inset) Illustration of the string pattern formation due to a hole moving through an AFM ordered state, which leaves behind a trail of displaced spins. (e) and (f) Same as in (c) and (d) but now the RUGAN is not provided with data corresponding to large doping values $\delta \gtrsim 0.24$ during training and is required to extrapolate to unseen values of $\delta$. For the extrapolation regime, we show the statistics obtained by two independently trained RUGANs.

Figure 3

(a) Decay of sign-corrected spin correlations $C_s\left(r\right)$ of Eq. (2) with temperature $T$ for nearest neighbors ($r=1$) for configurations obtained through experiment and with RUGAN. (b) Same as panel (a) but now for next-nearest neighbors ($r=\sqrt{2}$). (c–f) Snapshots generated at the temperatures indicated in panel (a). Each row is created with the same noise instance $\mathbf{z}$, and the marked sites indicate the change in a configuration upon increasing the temperature.

Figure 4

(a) Example snapshots of a doped Fermi-Hubbard model created at a scale that has approximately four times as many sites as the training examples (orange line) in Fig. 1. (b) The number of strings exceeding a length of 2 per system site (circles) and the average length (sites) of the strings (squares), for large-scale snapshots created with the RUGAN. For each value of $\delta$, we use the RUGAN of Figs. 2 and 2 to create 1000 large-scale snapshots.

Figure 5

Relative error on the average string length for RUGANs trained on different subsets of the data. For the green line, the doping values on which we train are cut off to only include the 12 lowest values [same as in Fig. 2]; this is 9 and 7 for the blue and orange line, respectively.

Figure 6

Sign-corrected spin correlations $C_s\left(r\right)$ as in Figs. 1 and 1, but now for (a) $r=2$ and (b) $r=4$.

Figure 7

(a) The average length (sites) of the string patterns, and (b) the number of strings exceeding a length of 2 per system site as a function of the doping $\delta$. The RUGAN results are obtained with 1000 generated spin configurations for each $\delta$. The doping values $\delta \lesssim 0.09$ in the shaded area are not included during training of the generative model.

Figure 8

(a) The setup of the RUGAN method. The generator $G$ (blue) transforms latent samples of size $l$, and optionally labels $\left\{\lambda \right\}$, to proposed configurations of size $4l$. The critic $C$ (pink) is used to measure the distance between the distributions of generated and real configurations. This information is used to update both neural networks. (b) The generator consists of translationally equivariant convolutional operations and therefore can be applied to inputs of arbitrary size, allowing for the creation of configurations at different spatial scales. (c) The $l\mathrm{th}$ hidden layer of the residual convolutional networks transforms an input ${\mathbf{x}}_l$ into ${\mathbf{x}}_{l+1}$, which can have a different spatial scale than the input. When reducing the spatial scale, the rescaling operation is an average-pooling layer with kernel size 2 and stride 2. When increasing the spatial scale, the rescaling consists of nearest-neighbor interpolation with a scale factor of 2. The convolutional operations with their corresponding kernel sizes are shown in blue. Note that we do not use batch normalization in the critic [48]. (d) Ising configurations created by a RUGAN, trained on $L=64$ configurations, at different spatial scales. The training size is shown in red on the larger configurations.

Figure 9

(a) During training of a RUGAN on classical Ising model data, we condition the RUGAN on the energy and magnetization of each configuration, and only show it examples with labels in the regions enclosed by dashed lines. After training, it is requested to create configurations with all possible labels; we show the median absolute error in the magnetization. The results are obtained by sampling 100 configurations at possible combinations of the energy and magnetization per site with energy spacing $\mathrm{\Delta }E/L^2=1/64$ and magnetization spacing $\mathrm{\Delta }m=1/1024$. Results shown here are for the training system size $L=64$. (b) Same as panel (a), but now showing the median absolute error in the energy. (c) Average error and its standard deviation at fixed energy [indicated by the blue line in panel (a)] made by the RUGAN for generating configurations with a requested value of the magnetization. (d) Average error and its standard deviation at fixed magnetization [indicated by the blue line in panel (b)] made by the RUGAN for generating configurations with a requested value of the energy. (e–h) Same as panels (a–d), but now with configurations created at a larger scale $L=256$, containing 16 times more spins than the training examples.

Figure 10

The distribution of configurations of an $L=10$ TFI created by RUGAN (blue) cannot be distinguished from the training examples with other unsupervised machine learning methods. Here, we show a dimensionality reduction with the UMAP algorithm [25] of an equal number of RUGAN and QMC configurations. Note that the RUGAN was not trained on $h=1.1$.